EVAT 554 OCEAN-ATMOSPHERE DYNAMICS FILTERING OF EQUATIONS FOR OCEAN LECTURE 10 (Reference: Peixoto & Oort, Chapter 3,8)
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EVAT 554OCEAN-ATMOSPHERE
DYNAMICS
FILTERING OF EQUATIONS FOR OCEAN
LECTURE 10
(Reference: Peixoto & Oort, Chapter 3,8)
Note that the major horizontal ocean
circulation systems mirror closely the semi-
permanent high and low pressure
systems
Scale AnalysisThe Ocean
)/()(/cosˆ1'v
/)(//
zuz
upa
wff
zuwutudtdu
VH
V
Zonal Momentum Balance:
)/v()v(/ˆ1
/vv)(/v/v
zz
pa
fu
zwtdtd
VH
VMeridional Momentum Balance:
Vertical Momentum Balance:
)/()w(/ˆ1
ˆ'
/)(//
zwz
zpguf
zwwwtwdtdw
VH
V
Continuity: 01 zw
dtd V
),,( TSpEquation of State:
dtdp
CpCplatq
Cpradq
zTz
T
zTwTtTdtdT
VH 1)/()(
/)(//
VHeat Equation:
(incompressible!)
How many equations?
How many variables?
)/()(/cosˆ1'v/ zu
zup
awffdtdu VH
Zonal Momentum Balance:
222 //ˆ/'v/ huluLpwffLuVH
Length scale: L106m, l105m
Depth scale: H103m, h 102m
Horizontal velocity scale: u,v 10-1 ms-1
Vertical velocity scale: w 10-4 ms-1
Horizontal pressure scale: p 100 mb = 104 Pa
Time Scale: L/u 107s H/w 107s
Radius of Earth: a=6.37x 106m
Coriolis parameter: f,f' 10-4 s-1
Density of Water: 1000 kg m-3
Horizontal Eddy Viscosity: H 105 m2s-1
Vertical Eddy Viscosity: V 10-1 m2s-1
10-8 ms-2
/cosˆ1v0 pa
f
10-5 ms-2 10-8 ms-2 10-5 ms-2 10-6 ms-2 10-6 ms-2
Meridional Momentum Balance:
10-8 ms-2 10-5 ms-2 10-5 ms-2 10-6 ms-2 10-6 ms-2
)/v()v(/ˆ1/v z
zp
afudtd VH
222 //ˆ/u/ huluLpfLuVH
/ˆ10 pa
fu
Length scale: L106m, l105m
Depth scale: H103m, h 102m
Horizontal velocity scale: u,v 10-1 ms-1
Vertical velocity scale: w 10-4 ms-1
Horizontal pressure scale: p 100 mb = 104 Pa
Time Scale: L/u 107s H/w 107s
Radius of Earth: a=6.37x 106m
Coriolis parameter: f,f' 10-4 s-1
Density of Water: 1000 kg m-3
Horizontal Eddy Viscosity: H 105 m2s-1
Vertical Eddy Viscosity: V 10-1 m2s-1
Horizontal Momentum Balance
10-8 ms-2 10-5 ms-2 10-5 ms-2 10-6 ms-2 10-6 ms-2
Geostrophic Balance
“Rossby Number”
Geostrophic Balance Holds when Ro << 1
(zonal) (meridional)
|u||/| 2
fLuRo
3
64
1
101010
10
Lfu||||
222 //ˆ/u/ huluLpfLuVH
/cosˆ1v pa
f
/ˆ1 pa
fu
Length scale: L106m, l105m
Depth scale: H103m, h 102m
Horizontal velocity scale: u,v 10-1 ms-1
Vertical velocity scale: w 10-4 ms-1
Horizontal pressure scale: p 100 mb = 104 Pa
Time Scale: L/u 107s H/w 107s
Radius of Earth: a=6.37x 106m
Coriolis parameter: f,f' 10-4 s-1
Density of Water: 1000 kg m-3
Horizontal Eddy Viscosity: H 105 m2s-1
Vertical Eddy Viscosity: V 10-1 m2s-1
Horizontal Momentum Balance
10-8 ms-2 10-5 ms-2 10-5 ms-2 10-6 ms-2 10-6 ms-2
Geostrophic Balance
Geostrophic Balance Holds when Ek << 1
(zonal) (meridional)
|u|
|/| 2
f
luEk H
H
1
104
5
101010
10
2|| lf
vH
222 //ˆ/u/ huluLpfLuVH
/cosˆ1v pa
f
/ˆ1 pa
fu
“Ekman Number”
Length scale: L106m, l105m
Depth scale: H103m, h 102m
Horizontal velocity scale: u,v 10-1 ms-1
Vertical velocity scale: w 10-4 ms-1
Horizontal pressure scale: p 100 mb = 104 Pa
Time Scale: L/u 107s H/w 107s
Radius of Earth: a=6.37x 106m
Coriolis parameter: f,f' 10-4 s-1
Density of Water: 1000 kg m-3
Horizontal Eddy Viscosity: H 105 m2s-1
Vertical Eddy Viscosity: V 10-1 m2s-1
Horizontal Momentum Balance
10-8 ms-2 10-5 ms-2 10-5 ms-2 10-6 ms-2 10-6 ms-2
Geostrophic Balance
Geostrophic Balance Holds when Ek << 1
(zonal) (meridional)
|u|
|/| 2
f
huEk V
V
1
44
1
101010
10
2|| hf
vV
222 //ˆ/u/ huluLpfLuVH
/cosˆ1v pa
f
/ˆ1 pa
fu
“Ekman Number”
Length scale: L106m, l105m
Depth scale: H103m, h 102m
Horizontal velocity scale: u,v 10-1 ms-1
Vertical velocity scale: w 10-4 ms-1
Horizontal pressure scale: p 100 mb = 104 Pa
Time Scale: L/u 107s H/w 107s
Radius of Earth: a=6.37x 106m
Coriolis parameter: f,f' 10-4 s-1
Density of Water: 1000 kg m-3
Horizontal Eddy Viscosity: H 105 m2s-1
Vertical Eddy Viscosity: V 10-1 m2s-1
Horizontal Momentum Balance
Geostrophic Balance
(zonal) (meridional)
/cosˆ1v pa
f
/ˆ1 pa
fu
Length scale: L106m, l105m
Depth scale: H103m, h 102m
Horizontal velocity scale: u,v 10-1 ms-1
Vertical velocity scale: w 10-4 ms-1
Horizontal pressure scale: p 100 mb = 104 Pa
Time Scale: L/u 107s H/w 107s
Radius of Earth: a=6.37x 106m
Coriolis parameter: f,f' 10-4 s-1
Density of Water: 1000 kg m-3
Horizontal Eddy Viscosity: H 105 m2s-1
Vertical Eddy Viscosity: V 10-1 m2s-1
110 1
HVEkEk110 3 Ro
Note that these approximations are only appropriate for “interior solutions” and will break down in boundary layers, where
horizontal or vertical shear are large!
Or near the equator!!
Horizontal Momentum Balance
Geostrophic Balance
(zonal) (meridional)
/cosˆ1v pa
f
/ˆ1 pa
fu
Horizontal Momentum Balance
Geostrophic Balance
(zonal) (meridional)
/cosˆ1v pa
f
/ˆ1 pa
fu
Dynamic Topography
Horizontal Momentum Balance
Geostrophic Balance
(zonal) (meridional)
/cosˆ1v pa
f
/ˆ1 pa
fu
Dynamic Topography
Horizontal Momentum Balance
Geostrophic Balance
(zonal) (meridional)
/cosˆ1v pa
f
/ˆ1 pa
fu
Dynamic Topography
the dynamic typography is not a simple consequence of
the overlying sea level pressure
requires an understanding of ocean
dynamics and its relation with atmospheric
windstress
22 //ˆ/u'/ HwLwHpgfLwuVHb
Vertical Momentum Balance:
Length scale: L106m, l105m
Depth scale: H103m, h 102m
Horizontal velocity scale: u,v 10-1 ms-1
Vertical velocity scale: w 10-4 ms-1
Horizontal pressure scale: p 100 mb = 104 Pa
Time Scale: L/u 107s H/w 107s
Radius of Earth: a=6.37x 106m
Coriolis parameter: f,f' 10-4 s-1
Density of Water: 1000 kg m-3
Horizontal Eddy Viscosity: H 105 m2s-1
Vertical Eddy Viscosity: V 10-1 m2s-1
10-11 ms-2 10-5 ms-2 10 ms-2 10 ms-2 10-11 ms-2 10-11 ms-2
)/()w(/ˆ1
ˆ'/ zw
zzpgufdtdw VH
zpg /ˆ1
ˆ0
fag
zu
cosv
fag
z
But now, depends on T,S,p
As with the atmosphere, we can combine geostrophic and hydrostatic balance to get
We can’t go proceed until we develop the equation of state for ocean water…
Thermal Wind Balance
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